847/845: Difference between revisions

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| Comma = yes

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'''847/845''', the '''cuthbert comma''', is a [[13-limit]] (also [[5.7.11.13 subgroup]]) [[comma]] measuring about 4.09{{cent}}. It is the difference between [[7/5]] and a stack of two [[13/11]]'s. It is also the difference between [[125/121]] and [[175/169]].

'''847/845''', the '''cuthbert comma''', is a [[small comma|small]] [[13-limit]] (also [[5.7.11.13 subgroup|5.7.11.13-subgroup]]) [[comma]] measuring about 4.09{{cent}}. It is the difference between [[7/5]] and a stack of two [[13/11]]'s. It is also the difference between [[125/121]] and [[175/169]].

In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, [[637/625]] is about as simple but much larger, and [[2941225/2924207]] is significantly more complex yet still twice as large.

In terms of full 13-limit commas, it is the difference between the following superparticular pairs:

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* [[385/384]] and [[4225/4224]]

Meanwhile, it can be factorized as [[1001/1000]] × [[2200/2197]] or [[441/440]] × [[10648/10647]].

Meanwhile, it can be factorized as ([[1001/1000]])⋅([[2200/2197]]) or ([[441/440]])⋅([[10648/10647]]).

In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, [[637/625]] is about as simple but much larger, and [[2941225/2924207]] is significantly more complex yet still twice as large.

== Temperaments ==

[[Tempering out]] this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the [[cuthbert chords]].

[[Tempering out]] this comma in the 13-limit results in the rank-5 '''cuthbert''' temperament and enables the [[cuthbert chords]].

Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament.

~~== See also ==~~

* [[Small comma]]

[[Category:Cuthbert]]

[[Category:Commas with unknown etymology]]

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
-'''847/845''', the '''cuthbert comma''', is a [[13-limit]] (also [[5.7.11.13 subgroup]]) [[comma]] measuring about 4.09{{cent}}. It is the difference between [[7/5]] and a stack of two [[13/11]]'s. It is also the difference between [[125/121]] and [[175/169]].
+'''847/845''', the '''cuthbert comma''', is a [[small comma|small]] [[13-limit]] (also [[5.7.11.13 subgroup|5.7.11.13-subgroup]]) [[comma]] measuring about 4.09{{cent}}. It is the difference between [[7/5]] and a stack of two [[13/11]]'s. It is also the difference between [[125/121]] and [[175/169]].
-In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, [[637/625]] is about as simple but much larger, and [[2941225/2924207]] is significantly more complex yet still twice as large.
 In terms of full 13-limit commas, it is the difference between the following superparticular pairs:
@@ Line 14: / Line 12: @@
 * [[385/384]] and [[4225/4224]]
-Meanwhile, it can be factorized as [[1001/1000]] × [[2200/2197]] or [[441/440]] × [[10648/10647]].
+Meanwhile, it can be factorized as ([[1001/1000]])⋅([[2200/2197]]) or ([[441/440]])⋅([[10648/10647]]).
+In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, [[637/625]] is about as simple but much larger, and [[2941225/2924207]] is significantly more complex yet still twice as large.
 == Temperaments ==
-[[Tempering out]] this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the [[cuthbert chords]].
+[[Tempering out]] this comma in the 13-limit results in the rank-5 '''cuthbert''' temperament and enables the [[cuthbert chords]].
 Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament.
-== See also ==
-* [[Small comma]]
 [[Category:Cuthbert]]
 [[Category:Commas with unknown etymology]]